1. Field of the Invention
This invention relates generally to the field of quantum cryptography, and more particularly to a method for exchanging a key with guaranteed security using systems vulnerable to photon number splitting (PNS) attacks, i.e. a quantum cryptography protocol robust against PNS attacks.
2. Discussion of Prior Art
If two users possess shared random secret information (below the “key”), they can achieve, with provable security, two of the goals of cryptography: 1) making their messages unintelligible to an eavesdropper and 2) distinguishing legitimate messages from forged or altered ones. A one-time pad cryptographic algorithm achieves the first goal, while Wegman-Carter authentication achieves the second one. Unfortunately both of these cryptographic schemes consume key material and render it unfit for use. It is thus necessary for the two parties wishing to protect the messages they exchange with either or both of these cryptographic techniques to devise a way to exchange fresh key material. The first possibility is for one party to generate the key and to inscribe it on a physical medium (disc, cd-rom, rom) before passing it to the second party. The problem with this approach is that the security of the key depends on the fact that it has been protected during its entire lifetime, from its generation to its use, until it is finally discarded. In addition, it is very unpractical and tedious.
Because of these difficulties, in many applications one resorts instead to purely mathematical methods allowing two parties to agree on a shared secret over an insecure communication channel. Unfortunately, all such mathematical methods for key agreement rest upon unproven assumptions, such as the difficulty of factoring large integers. Their security is thus only conditional and questionable. Future mathematical developments may prove them totally insecure.
Quantum cryptography (QC) is the only method allowing the distribution of a secret key between two distant parties, the emitter and the receiver, [1] with a provable absolute security. Both parties encode the key on elementary quantum systems, such as photons, which they exchange over a quantum channel, such as an optical fiber. The security of this method comes from the well-known fact that the measurement of an unknown quantum state modifies the state itself: a spy eavesdropping on the quantum channel cannot get information on the key without introducing errors in the key exchanged between the emitter and the receiver. In equivalent terms, QC is secure because of the no-cloning theorem of quantum mechanics: a spy cannot duplicate the transmitted quantum system and forward a perfect copy to the receiver.
Several QC protocols exist. These protocols describe how the bit values are encoded on quantum states and how the emitter and the receiver cooperate to produce a secret key. The most commonly used of these protocols, which was also the first one to be invented, is known as the Bennett-Brassard 84 protocol (BB84) [2]. The emitter encodes each bit on a two-level quantum system either as an eigenstate of σx (|+x coding for “0” and |−x coding for “1”) or as an eigenstate of σy (|+y or |−y, with the same convention). The quantum system is sent to the receiver, who measures either σx or σy. After the exchange of a large number of quantum systems, the emitter and the receiver perform a procedure called basis reconciliation. The emitter announces to the receiver, over a conventional and public communication channel the basis x or y (eigenstate of σx or σy) in which each quantum system was prepared. When the receiver has used the same basis as the emitter for his measurement, he knows that the bit value he has measured must be the one which was sent over by the emitter. He indicates publicly for which quantum systems this condition is fulfilled. Measurements for which the wrong basis was used are simply discarded. In the absence of a spy, the sequence of bits shared is error free. Although a spy who wants to get some information about the sequence of bits that is being exchanged can choose between several attacks, the laws of quantum physics guarantee that he will not be able to do so without introducing a noticeable perturbation in the key.
Other protocols—like the Bennett 92 (B92) [3]—have been proposed.
In practice, the apparatuses are imperfect and also introduce some errors in the bit sequence. In order to still allow the production of a secret key, the basis reconciliation part of the protocol is complemented by other steps. This whole procedure is called key distillation. The emitter and the receiver check the perturbation level, also know as quantum bit error rate (QBER), on a sample of the bit sequence in order to assess the secrecy of the transmission. In principle, errors should be encountered only in the presence of an eavesdropper. In practice however, because of the imperfections of the apparatus, a non-zero error probability can also always be observed. Provided this probability is not too large, it does not prevent the distillation of a secure key. These errors can indeed be corrected, before the two parties apply a so called privacy amplification algorithm that will reduce the information quantity of the spy to an arbitrarily small level.
In the last years, several demonstrations of QC systems have been implemented using photons as the information carriers and optical fibers as quantum channels. While the original proposal called for the use of single photons as elementary quantum systems to encode the key, their generation is difficult and good single-photon sources do not exist yet. Instead, most implementations have relied on the exchange between the emitter and the receiver of weak coherent states, such as weak laser pulses, as approximations to ideal elementary quantum systems. Each pulse is a priori in a coherent state |μeiθ of weak intensity (typically the average photon number per pulse μ≈0.1 photons). However since the phase reference of the emitter is not available to the receiver or the spy, they see a mixed state, which can be re-written as a mixture of Fock states,
            ∑      n        ⁢                  ⁢                  p        n            ⁢                      n        〉            ⁢              〈        n                      ,where the number n of photons is distributed according to Poissonian statistics with mean μ and pn=e−μμn/n!. QC with weak pulses can be re-interpreted as follows: a fraction p1 of the pulses sent by the emitter contain exactly one photon, a fraction p2 two photons, and so on, while a fraction p0 of the pulses are simply empty and do not contribute to the key transmission. Consequently, in QC apparatuses employing weak pulses, a rather important fraction of the non-empty pulses actually contain more than one photon. The spy is then not limited any longer by the no-cloning theorem. He can simply keep some of the photons while letting the others go to the receiver. Such an attack is called photon-number splitting (PNS) attack. If we assume that the only constraints limiting the technological power of the spy are the laws of physics, the following attack is in principle possible: (1) for each pulse, the spy counts the number of photons, using a photon number quantum non-demolition measurement; (2) he blocks the single photon pulses, while keeping one photon of the multi-photon pulses in a quantum memory and forwarding the remaining photons to the receiver using a perfectly transparent quantum channel; (3) he waits until the emitter and the receiver publicly reveal the bases used, and correspondingly measures the photons stored in his quantum memory: he must discriminate between two orthogonal states, and this can be done deterministically. In this way, he obtains full information on the key, which implies that no procedure allows to distillate a secret key for the legitimate users. In addition, the spy does not introduce any discrepancies in the bit sequences of the emitter and the receiver. The only constraint on PNS attacks is that the presence of the spy should remain undetected. In particular, he must ensure that the rate of photons received by the receiver is not modified.
In the absence of the spy, the raw rate of photons that reach the receiver is given by:RReceiver(δ)=μ·10−δ/10 [photons/pulse]  (1)where δ=α L is the total attenuation in dB of the quantum channel of length L. Thus, the PNS attack can be performed on all passing pulses only when δ≧δc with RReceiver(δc)≅p2: the losses that the receiver expects because of the fiber attenuation are equal to those introduced by the action of the spy storing and blocking photons. For shorter distances, the spy sends a fraction q of the pulses on her perfectly transparent channel without doing anything and performs the PNS attack on the remaining 1−q fraction of the pulses. The receiver measures a raw detection rateRReceiver|Spy(q)=qμ+(1−q)B[photons/pulse]  (2)where
  B  =            ∑              n        ⁢                  >          _                ⁢        2              ⁢                  ⁢                            p          n                ⁡                  (                      n            -            1                    )                    .      The parameter q is chosen so that RReceiver|spy(q)=RReceiver(δ). The information the spy gets on a bit sent by the emitter is 0 when he does nothing, and 1 when he perform the PNS attack, provided of course that the receiver has received at least one photon:
                                          I            Spy                    ⁡                      (            q            )                          =                                                            (                                  1                  -                  q                                )                            ⁢              S                                      q              +                                                (                                      1                    -                    q                                    )                                ⁢                S                                              ⁢                                          [                      bits            ⁢                          /                        ⁢            pulse                    ]                                    (        3        )            with
  S  =            ∑              n        ⁢                  >          _                ⁢        2              ⁢                  ⁢                  p        n            .      The critical length of the quantum channel is determined by the condition RReceiver(δc)=RReceiver|Spy (q=0). For an average photon number μ=0.1, one finds δc=13 [dB], which corresponds to a distance of the order of 50 km (α=0.25 [dB/km])
Although the PNS attacks are far beyond today's technology, their consequences on the security of a QC system relying on weak coherent states is devastating, when they are included in the security analysis [4]. The extreme vulnerability of the BB84 protocol to PNS attacks is due to the fact that whenever the spy can keep one photon, he gets all the information, since he has to discriminate between two eigenstates of a known Hermitian operator, which is allowed by the laws of quantum physics.